Risk Premia and How Sure You Can Be

Every one of these has a positive average. That is not the same as being reliable, and the difference is measurable.

Worked example — Risk Premia and How Sure You Can Be

Average per year	8.44%
Standard deviation	19.70%
t-statistic	4.24
Years it paid	66/98

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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A century of data is a small sample 🖖

The size premium averages 5.92% a year over 98 years and is positive in 49 of them. A coin flip that pays 5.92% on average is still a coin flip, and the standard deviation of 27.48% is why: to tell that average from zero at two standard errors you would need roughly 87 years, and the term premium would need about 118 — more history than the data contains. This is not an argument that premia are fake. It is an argument that the confidence usually attached to them is borrowed from the size of the average rather than earned from the size of the sample.

A bigger premium can be the less certain one 🖖

How much history a premium needs depends only on the ratio of its noise to its size: put the average two standard errors from zero and the requirement is n ≥ (2σ/μ)². Size and certainty are therefore separate questions, and on this record they disagree. The credit premium averages 2.08% a year and needs about 39 years; the size premium averages nearly three times as much at 5.92% and needs 87. Step through the four and the requirement runs 22, 39, 87, 118 years while the averages run 8.44%, 2.08%, 5.92%, 1.41% — the same four premia in two orders that do not match.

The size premium was a 1928–1976 phenomenon 🖖

Split the record in half with the start and span fields. Over 1928–1976 small US companies beat large ones by 10.70% a year; over 1977–2025 the same premium was 1.14%, with a t-statistic of 0.42 and a sample requirement of 1,133 years. The 5.92% headline is the average of a large first half and almost nothing since. Banz measured the effect in 1981, near the join. This tool cannot tell you whether publishing it competed the premium away or whether the first half was the accident, and neither can anyone else — but it can tell you that a 98-year average was never a description of both halves.

Problem solved in full

An equity premium of 8.44% and a 19.70% standard deviation 6 steps
Three numbers off the cards settle how sure the equity premium is: 8.44% a year, a 19.70% standard deviation, and 98 years. Get the certainty out of them — then ask the question backwards, and find how much history a premium of this shape would need. This is the tool's opening state: Equity (S&P 500 − 3-month bill) over the whole record from 1928.
1. Nothing else about the market enters below. Not the level of prices, not the number of crashes, not which decade you happened to start in — a mean, a spread and a count.
  
  $μ = 8.44%, σ = 19.70%, n = 98$
2. One year is wildly uncertain; the average of 98 of them is not, and the improvement is exactly the square root of the count. Divide, and the panel How much of this is measurement, and how much is noise has the same figure.
  
  $SE = \frac{σ}{n} = \frac{19.70%}{9.8995} = 1.99%$
3. The t-statistic is that average re-expressed in units of its own uncertainty, so the equity premium sits four standard errors clear of zero. Notice which number did the work: not 8.44% being large, but 98 years shrinking a 19.70% wobble down to 1.99%.
  
  $t = \frac{μ}{SE} = \frac{8.44}{1.99} = 4.24$
4. Now run it backwards. Fix the bar at two standard errors and solve for n instead, and the years needed depend on the ratio σ/μ alone. So a bigger premium is not automatically a surer one: double the average and the spread together and the requirement does not move. The same panel prints 22.
  
  $t = 2 : n_{m i n} = (\frac{2 σ}{μ})^{2} = (\frac{2 \times 19.70}{8.44})^{2} = 4.668 2^{2} = 21.79 \to 22$
5. Choose Term (10-year bond − 3-month bill) in the Premium dropdown. The average falls to 1.41% and the spread to 7.63%, and the requirement jumps to 118 years against a record of 98. Term misses the bar not because 1.41% is small, but because 7.63% is large beside it.
  
  $μ = 1.41%, σ = 7.63% : n_{m i n} = (\frac{2 \times 7.63}{1.41})^{2} = 10.822 7^{2} = 117.13 \to 118 > 98$
6. Last, stop solving for n. Ninety-eight years is not a choice — it is the whole annual record — so fix it and solve for the smallest average that could clear the bar. At equity-like volatility it is simply twice the standard error.
  
  $n = 98 : μ_{m i n} = \frac{2 σ}{n} = 2 SE = 2 \times 1.99% = 3.98%$
Answer

A 1.99% standard error, t = 4.24, and 22 years would have been enough — but with a 19.70% spread and 98 years of data, no premium averaging under 3.98% a year could ever be told from zero. That floor belongs to the archive rather than to the strategy, and every premium on the dropdown carries its own version of it: 2σ/√98 is 5.55% for size, 1.54% for term, 1.30% for credit. Set those beside the averages the tool reports for each — 5.92%, 1.41%, 2.08% — and you have called all four verdicts without computing a single t-statistic, term being the only one that lands below its own floor, and by 0.13 of a point. Waiting does almost nothing, because the floor falls as √n: ten more years of data moves the equity floor from 3.98% to 3.79%, and halving it to 1.99% would take 392 years. The tool computes how many years a premium of a given size would need. It never computes the smallest size this record could ever see, which is the number that decides in advance which arguments about premia are arguments about markets and which are arguments about noise.

$n_{m i n} = (\frac{2 σ}{μ})^{2} ⟺ μ_{m i n} = \frac{2 σ}{n} = \frac{2 \times 19.70%}{9.8995} = 3.98%$

Learning path

Risk, measured

Leads to The frontier, held the size of the error bar around a long-run average return.

References (3)

The equity premium stated as a problem rather than a fact — it is too large for standard theory to explain, which is itself a reason to hold the estimate loosely: R. Mehra and E. C. Prescott, "The equity premium: A puzzle." Journal of Monetary Economics 15(2), 145–161, 1985.
The size effect, in the paper that first measured it: R. W. Banz, "The relationship between return and market value of common stocks." Journal of Financial Economics 9(1), 3–18, 1981.
And the standing warning that a mean return is measured far less precisely than it is quoted: R. C. Merton, "On estimating the expected return on the market: An exploratory investigation." Journal of Financial Economics 8(4), 323–361, 1980.

By the 42Math team · Updated 2026-08-19

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Equity, whole record - The equity premium over 1928-2025: the strongest of the four, and still negative in a third of the years.
Size, whole record - The size premium over the whole record — a large average that was positive in only about half the years.
Term, whole record - The term premium, whose t-statistic stays under 2 even on 98 years of data.
Equity since 1990 - The equity premium since 1990, a shorter window and correspondingly less certain.